Electromagnetic Waves

Transmission of energy through a vacuum or using no medium is

accomplished by electromagnetic waves, caused by theosscilation of electric and magnetic fields. They move at a constantspeed of 3x108 m/s. Often, they are called electromagneticradiation, light, or photons.

Did you ever wonder what is electromagnetic radiation? The word

is somewhat complicated, but you are in contact withelectromagnetic radiation all the time. Here is a diagram of theelectromagnetic radiation spectrum that has appeared in many textbooks and websites. Electromagnetic radiation is caused by the disturbance of anelectromagnetic field.

The last line of numbers in power of 10 gives the wavelength in m. The regions sometimes donot have a clear cut, because there is considerable overlap. For example, radio waves andmicrowaves bondary is very vague, but public regulation for their application (usage) is strict.

Electromagnetic waves are used to transmit long/short/FM wavelength radio waves, andTV/telephone/wireless signals or energies. They are also responsible for transmiting energy inthe form of microwaves, infrared radiation (IR), visible light (VIS), ultraviolet light (UV), X-rays, and gamma rays. Each region of this spectrum plays an important part in our lives, and inthe business involving communication technology. The list given above are in increasingfrequency (or decreasing wavelength) order. Here again is the list of regions and the approximatewavelengths in them. For simplicity, we choose to give only the magnitudes of frequencies. Thatis we give log (frequency) (log(f)).

Electromagnetic radiations are usually treated as wave motions. The electronic and magneticfields oscillate in directions perpendicular to each other and to the direction of motion of thewave.The wavelength, the frequency, and the speed of light obey the following relationship:

wavelength * frequency = speed of light.

The speed of light is usually represented by c, the wavelength by the lower case Greek letterlambda,  and the frequency by lower case Greek letter nu . In these symbols, the aboveformula is:

 = c

The electromagnetic radiation is the fundation for radar, which is used for guidance and remotesensing for the study of the planet Earth.

The Visible Spectrum

Wavelengths of the visible region of the spectrum range from 700 nm for red light to 400 nm forviolet light.

red 700 nm orange 630 yellow 550 green 500 blue 450 violet 400There is no need to memorize these numbers, but knowing that the visible region has such anarrow range of 400-700 nm is handy at times when referring to certain light.

Photons - bundles of electromagnetic energy

In his research on the radiation from a hot (black) body, Max. Planck made a simple proposal.He suggested that light consists of photons. The energy, E, of each individual photon of amonochromatic light wave, is proportional to the frequency, , of the light:

E=hwhere h (= 6.626*10-34 J s) is now known as the Planck constant. Often, we write h = 6.626e-34J s for simplicity.

For the convenience of your future study of electromagnetic radiation, you might want to knowthe units often used for it.

1 Hz = 1, hertz: cycle per second, for frequency

When photons shines on a metal plate, they free electrons. Energy is

required to pull the electrons out of the metal surface, and thisamount of energy is called threshold. The excess energy of thephoton appear as the kinetic energy of the electrons. Here is asimulation of, which gives a slightly different perspective. Athe photoelectric effectphotoelectric effect demonstration is also funfor you if you have the time.

Einstein learned of Planck's proposal, and he wanted to perform

experiments to show if the proposal is true. At that time, thephotoelectric effect was known, and he measured the kinetic energyof electrons released by photons. He did find a linear relationshipbetween the kinetic energy of the electrons and the frequency of lightused, (see diagram below).

Furthermore, he found the light of minimum frequency needed to

release electrons from a metal to be constant, and this energy must beovercome in order to take the electron out of the metal. This energy iscalled the threshold energy, W. The formula to descirbe photoelectronkinetic energy Ek is

Ek = h v – W

and the proportional constant is what is known as the Planck Constant.

The minimum frequency is called threshold frequency. Thequantity h v is the energy of the photon. In other words, the energy ofthe photon is completely given to the electron:

h v = Ek + W

Energy is conserved, neither created, nor destroyed. This formula also illustrates the (great)principle of conservation of energy.Maxwell's EquationsThe Equations

Maxwell’s four equations describe the electric and magnetic fields arising from distributions ofelectric charges and currents, and how those fields change in time. They were the mathematicaldistillation of decades of experimental observations of the electric and magnetic effects ofcharges and currents, plus the profound intuition of Michael Faraday. Maxwell’s owncontribution to these equations is just the last term of the last equation—but the addition of thatterm had dramatic consequences. It made evident for the first time that varying electric andmagnetic fields could feed off each other—these fields could propagate indefinitely throughspace, far from the varying charges and currents where they originated. Previously these fieldshad been envisioned as tethered to the charges and currents giving rise to them. Maxwell’s newterm (called the displacement current) freed them to move through space in a self-sustainingfashion, and even predicted their velocity—it was the velocity of light!

Here are the equations:

1. Gauss’ Law for electric fields:

∫E→⋅dA→=q/ε0.

(The integral of the outgoing electric field over an area enclosing a volume equals the totalcharge inside, in appropriate units.)

2. The corresponding formula for magnetic fields:

∫B→⋅dA→=0.

(No magnetic charge exists: no “monopoles”.)

3. Faraday’s Law of Magnetic Induction:

∮E→⋅dℓ→=−d/dt(∫B→⋅dA→).

The first term is integrated round a closed line, usually a wire, and gives the total voltage changearound the circuit, which is generated by a varying magnetic field threading through the circuit.4. Ampere’s Law plus Maxwell’s displacement current:

∮B→⋅dℓ→=μ0(I+ddt(ε0∫E→⋅dA→)).

This gives the total magnetic force around a circuit in terms of the current through the circuit,plus any varying electric field through the circuit (that’s the “displacement current”).

The purpose of this lecture is to review the first three equations and the original Ampere’s lawfairly briefly, as they were covered earlier in the course, then to demonstrate why thedisplacement current term must be added for consistency, and finally to show, without usingdifferential equations, how measured values of static electrical and magnetic attraction aresufficient to determine the speed of light.

Preliminaries: Definitions of µ0 and ε0

Ampere discovered that two parallel wires carrying electric currents in the same direction attracteach other magnetically, the force in newtons per unit length being given by

F=2(μ04π)I1I2r,

for long wires a distance r apart. We are using the standard modern units (SI). Theconstant μ0/4π that appears here is exactly 10-7, this defines our present unit of current,the ampere. To repeat: μ0/4π is not something to measure experimentally, it's just a funny wayof writing the number 10-7! That's not quite fair—it has dimensions to ensure that both sides ofthe above equation have the same dimensionality. (Of course, there's a historical reason for thisstrange convention, as we shall see later). Anyway, if we bear in mind that dimensions havebeen taken care of, and just write the equation

F=2⋅10−7⋅I1I2r,

It's clear that this defines the unit current—one ampere—as that current in a long straight wirewhich exerts a magnetic force of 2×10−7 newtons per meter of wire on a parallel wire one meteraway carrying the same current.

However, after we have established our unit of current—the ampere—we have also therebydefined our unit of charge, since current is a flow of charge, and the unit of charge must be theamount carried past a fixed point in unit time by unit current. Therefore, our unit of charge—the coulomb—is defined by stating that a one-amp current in a wire carries one coulomb persecond past a fixed point.

To be consistent, we must do electrostatics using this same unit of charge. Now, the electrostaticforce between two charges is (1/4πε0)q1q2/r2. The constant appearing here, nowwritten 1/4πε0, must be experimentally measured—its value turns out to be 9×109.

To summarize: to find the value of 1/4πε0, two experiments have to be performed. We mustfirst establish the unit of charge from the unit of current by measuring the magnetic forcebetween two current-carrying parallel wires. Second, we must find the electrostatic forcebetween measured charges. (We could, alternatively, have defined some other unit of currentfrom the start, then we would have had to find both μ0 and ε0 by experiments on magnetic andelectrostatic attraction. In fact, the ampere was originally defined as the current that deposited adefinite weight of silver per hour in an electrolytic cell).Heinrich Hertz In a series of brilliant experiments Heinrich Hertz discovered radio waves and established that James Clerk Maxwell’s theory of electromagnetism is correct.

Hertz also discovered the photoelectric effect, providing

one of the first clues to the existence of the quantum world. The unit of frequency, the hertz, is named in his honor.

Beginnings Heinrich Rudolf Hertz was born on February 22, 1857 in the German port city of Hamburg. He was the firstborn of five children.

His mother was Anna Elisabeth Pfefferkorn, the daughter

of a physician.

His father was Gustav Ferdinand Hertz, an attorney who became a Senator.

His paternal grandfather, a wealthy Jewish businessman, had married into a Lutheran family andconverted to Christianity.

Both of Heinrich’s parents were Lutherans, and he was raised in this faith. His parents, however,were more interested in his education than his religious status.

SchoolAged six, Heinrich began at the Dr. Wichard Lange School in Hamburg. This was a privateschool for boys run by the famous educator Friedrich Wichard Lange. The school operatedwithout religious influence; it used child-centered teaching methods, taking account of students’individual differences. It was also strict; the students were expected to work hard and competewith one another to be top of the class. Heinrich enjoyed his time at school, and indeed was topof his class.

Unusually, Dr. Lange’s school did not teach Greek and Latin – theclassics – needed for university entry. The very young Heinrich toldhis parents he wanted to become an engineer. When they looked for aschool for him, they decided that Dr. Lange’s alternative focus, whichincluded the sciences, was the best option.

Heinrich Hertz, aged about 12, with his father, mother, and two younger brothers.Heinrich’s mother was especially passionate about his education.Realizing he had a natural talent for making things and for drawing,she arranged draftsmanship lessons for him on Sundays at a technical college. He started theseaged 11.

Homeschool and Building Scientific Apparatus

Aged 15, Heinrich left Dr. Lange’s school to be educated at home. He had decided that perhapshe would like to go to university after all. Now he received tutoring in Greek and Latin toprepare him for the exams.

He excelled at languages, a gift he seems to have inherited from his father.

Professor Redslob, a language specialist who gave Heinrich some tuition in Arabic, advised hisfather that Heinrich should become a student of oriental languages. Never before had he metanyone with greater natural talent.

Heinrich also began studying the sciences and mathematics at home, again with the help of aprivate tutor.

He had a colossal appetite for hard work. His mother said:

“When he sat with his books nothing could disturb him or draw him away from them.”

Although he had left his normal school, he continued attending the technical college on Sundaymornings.

In the evenings he worked with his hands. He learned to operate a lathe. He built models andbegan constructing increasingly sophisticated scientific apparatus such as a spectroscope. Heused this apparatus to do his own physics and chemistry experiments.

Meanwhile, his interest in mathematics and physics continued to grow.

Becoming a ScientistPhysics in MunichAfter completing his army service, the 20-year-old Hertz moved to Munich to begin anengineering course in October 1877. A month later, after much internal anguish, he dropped outof the course. He had decided that above all else he wanted to become a physicist.

He enrolled at the University of Munich, choosing courses in advanced mathematics and

mechanics, experimental physics, and experimental chemistry.

After a successful year at Munich he moved to the University of Berlin because it had betterphysics laboratories than Munich.

Berlin, Helmholtz, and Recognition

In Berlin, aged 21, Hertz began working in the laboratories of the great physicist Hermann vonHelmholtz.Helmholtz must have recognized a rare talent in Hertz, immediately asking him to work on aproblem whose solution he was particularly interested in. The problem was the subject of a fiercedebate between Helmholtz and another physicist by the name of Wilhelm Weber.

The University of Berlin’s Philosophy Department, with Helmholtz’s encouragement, had

offered a prize to anyone who could solve the problem: Does electricity move with inertia?Alternatively, we could frame the question in the form: Does electric current have mass? Or, asframed by Hertz: Does electric current have kinetic energy?

He personally designed experiments which he thought would answer Helmholtz’s question. He

began to really enjoy himself, writing home:

“I cannot tell you how much more satisfaction it gives me to gain knowledge for myself and for others directly from nature, rather than to be merely learning from others and myself alone.” --HEINRICH HERTZ 1878

Doctor of PhysicsHertz declined this project; he believed the attempt, with no guarantee of success, would takeseveral years of work. He was ambitious and wanted to publish new results quickly to establishhis reputation.

Instead of working for the prize, he carried out a masterful three-month project onelectromagnetic induction. He wrote this up as a thesis. In February 1880, at the age of 23, histhesis brought him the award of a doctorate in physics. Helmholtz quickly appointed him as anassistant professor. Later that year Hertz wrote:

“I grow increasingly aware, and in more ways than expected, that I am at the center of my ownfield; and whether it be folly or wisdom, it is a very pleasant feeling.” --HEINRICH HERTZ

The Discovery of Radio Waves

If you would like a somewhat more detailed technical account of Hertz’s discovery of radiowaves, we have one here.

Well-Equipped Laboratories and Attacking the Greatest Problem

In March 1885, desperate to return to experimental physics, Hertz moved to the University ofKarlsruhe. Aged 28, he had secured a full professorship. He was actually offered two other fullprofessorships, a sign of his flourishing reputation. He chose Karlsruhe because it had the bestlaboratory facilities.

Wondering about which direction his research should take, his thoughts drifted to the prize workHelmholtz had failed to persuade him to do six years earlier: proving Maxwell’s theory byexperiment.Hertz decided that this mighty undertaking would be the focus of his research at Karlsruhe.

A Spark that Changed Everything

After some months of experimental trials, the apparently unbreakable walls that had frustrated allattempts to prove Maxwell’s theory began crumbling.

It started with a spark.

It started with a chance observation early in October 1886, when Hertz was showing students electric sparks. Hertz began thinking deeply about sparks and their effects in electric circuits. He began a series of experiments, generating sparks in different ways.He discovered something amazing. Sparks produced a regular electrical vibration within theelectric wires they jumped between. The vibration moved back and forth more often everysecond than anything Hertz had ever encountered before in his electrical work.

He knew the vibration was made up of rapidly accelerating and decelerating electric charges. IfMaxwell’s theory were right, these charges would radiate electromagnetic waves which wouldpass through air just as light does.

Producing and Detecting Radio Waves

In November 1886 Hertz constructed the apparatus shown below.

The Oscillator. At the ends are two hollow zinc spheres of diameter 30 cm. The spheres are each connected to copper wires which run into the middle where there is a gap for sparks to jump between.He applied high voltage a.c. electricity across the central spark-gap, creating sparks.

The sparks caused violent pulses of electric current within the copper wires. These pulsesreverberated within the wires, surging back and forth at a rate of roughly 100 million per second.

As Maxwell had predicted, the oscillating electric charges produced electromagnetic waves –radio waves – which spread out through the air around the wires. Some of the waves reached aloop of copper wire 1.5 meters away, producing surges of electric current within it. These surgescaused sparks to jump across a spark-gap in the loop.

This was an experimental triumph. Hertz had produced and detected radio waves. He had passedelectrical energy through the air from one device to another one located over a meter away. Noconnecting wires were needed.Taking it FurtherOver the next three years, in a series of brilliant experiments, Hertz fully verified Maxwell’stheory. He proved beyond doubt that his apparatus was producing electromagnetic waves,demonstrating that the energy radiating from his electrical oscillators could be reflected,refracted, produce interference patterns, and produce standing waves just like light.

Hertz’s experiment’s proved that radio waves and light waves were part of the same family,which today we call the electromagnetic spectrum.

The electromagnetic spectrum. Hertz discovered the radio part of the spectrum.Strangely, though, Hertz did not appreciate the monumental practical importance of theelectromagnetic waves he had produced.

“I do not think that the wireless waves I have discovered will have any practical application.”--HEINRICH HERTZ 1890This was because Hertz was one of the purest of pure scientists. He was interested only indesigning experiments to entice Nature to reveal its mysteries to him. Once he had achieved this,he would move on, leaving any practical applications for others to exploit.

The waves Hertz first generated in November 1886 quickly changed the world.

By 1896 Guglielmo Marconi had applied for a patent for wireless communications. By 1901 hehad transmitted a wireless signal across the Atlantic Ocean from Britain to Canada.

Hertz’s discovery was the foundation stone for much of our modern communications technology.Radio, television, satellite communications, and mobile phones all rely on it. Even microwaveovens use electromagnetic waves: the waves penetrate the food, heating it quickly from theinside.

Our ability to detect radio waves has also transformed the science of astronomy. Radioastronomy has allowed us to ‘see’ features we can’t see in the visible part of the spectrum. Andbecause lightning emits radio waves, we can even listen to lightning storms on Jupiter andSaturn.

Scientists and non-scientists alike owe a lot to Heinrich Hertz.

The Production of Electro Magnetic waves  A charged particle produces an electric field. This electric field exerts a force on other charged particles. Positive charges accelerate in the direction of the field and negative charges accelerate in a direction opposite to the direction of the field.  A moving charged particle produces a magnetic field. This magnetic field exerts a force on other moving charges. The force on these charges is always perpendicular to the direction of their velocity and therefore only changes the direction of the velocity, not the speed.  An accelerating charged particle produces an electromagnetic (EM) wave. Electromagnetic waves are electric and magnetic fields traveling through empty space with the speed of light c. A charged particle oscillating about an equilibrium position is an accelerating charged particle. If its frequency of oscillation is f, then it produces an electromagnetic wave with frequency f. The wavelength λ of this wave is given by λ = c/f. Electromagnetic waves transport energy through space. This energy can be delivered to charged particles a large distance away from the source.

Assume a charge q located near the origin is accelerating. It therefore produces electromagneticradiation. At some position r in space and at some time t, the electric field of theelectromagnetic wave produced by the accelerating charge is given by

Erad(r,t) = -[1/(4πε0)]*[q/(c2r')]*aperp(t - r'/c).

Let us analyze this expression. The electric field is proportional to the charge q. The bigger the accelerating charge, the bigger is the field. It decreases as the inverse of the distance r', which is the distance between the accelerating charge and the position where the field is observed. But it is not the distance at the time the field is observed, but the distance at some earlier time, called the retarded time, when the radiation field was produced. All electromagnetic waves travel with the speed of light c = 3*108 m/s. It takes them a time interval ∆t = ∆r/c to travel a distance ∆r. The electric field is also proportional to the acceleration of the charge. The larger the acceleration, the larger is thefield. In the above expression Erad(r,t) is proportional to aperp, the component of the accelerationperpendicular to the line of sight between r and the retarded position of the charge. Thedirection of Erad(r,t) is perpendicular to to this line of sight and its magnitude is proportional tothe component of the acceleration perpendicular to this line of sight.The figure on the right illustrates that point. The electric field is zero along a line of sight in thedirection of the acceleration, largest along a line of sight perpendicular to the direction of theacceleration, and always perpendicular to the line of sight.

The magnitude of aperp is a*sinθ, and the magnitude of the radiation field therefore isErad(r,t) = -[1/(4πε0)]*[q/(c2r)]*sinθ*a(t - r/c).Here θ is the angle between the line of sight and the direction of the acceleration.

The magnetic field of the electromagnetic wave is

perpendicular to the electric field and has magnitude Brad =Erad/c. For electromagnetic waves E and B are alwaysperpendicular to each other and perpendicular to thedirection of propagation. The direction of propagation isthe direction of E × B.

The radiation field Erad decreases as 1/r, while the static

Coulomb field decreases as 1/r2. The static field decreaseswith distance much faster than the radiation field, andtherefore the radiation field will dominate at large distance for accelerating chargedistributions. In addition, radiation fields are often produced by accelerating electrons, while thestatic fields are produced by all charges (positive nuclei and negative electrons) and cancel eachother out.

Far from the source of an electromagnetic wave, we often treat the EM wave as a plane wave. Asinusoidal plane EM wave traveling in the x-direction is of the form

E(x,t) = Emaxsin(kx - ωt + φ),

B(x,t) = Bmaxsin(kx - ωt + φ).

If, for a wave traveling in the x-direction, E points in

the y-direction, then B points in the z- direction. Electromagnetic waves are transverse waves.

The wave vector k points into the direction of

propagation, and its magnitude k = 2π/λ, where λ is the wavelength of the wave. The frequency f of thewave is f = ω/2π, ω is the angular frequency. The speed of any sinusoidal wave is the product ofits wavelength and frequency.

v = λf.The speed of any electromagnetic waves in free space is the speed of light c = 3*108 m/s.Electromagnetic waves in free space can have any wavelength λ or frequency f as long as λf = c.Visible light is any electromagnetic wave with wavelength λ between approximately 400 nm and750 nm. Physics Ma’am Aileen Rovillos